Abstract.
We propose a new definition of pre-image
entropy for continuous maps on noncompact topological spaces,
investigate fundamental properties of the new pre-image entropy, and
compare the new pre-image entropy with the existing ones. The
defined pre-image entropy generates that of Cheng and Newhouse. Yet,
it holds various basic properties of Cheng and Newhouse's pre-image
entropy, for example, the pre-image entropy of a subsystem is
bounded by that of the original system, topologically conjugated
systems have the same pre-image entropy, the pre-image entropy of the
induced hyperspace system is larger than or equal to that of the
original system, and in particular this new pre-image entropy
coincides with Cheng and Newhouse's pre-image entropy for compact
systems.